**Mathematical Surveys and Monographs**

Volume: 216;
2017;
430 pp;
Hardcover

MSC: Primary 30;

Print ISBN: 978-0-8218-4360-4

Product Code: SURV/216

List Price: $116.00

AMS Member Price: $92.80

MAA Member Price: $104.40

**Electronic ISBN: 978-1-4704-4046-6
Product Code: SURV/216.E**

List Price: $116.00

AMS Member Price: $92.80

MAA Member Price: $104.40

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#### Supplemental Materials

# An Introduction to the Theory of Higher-Dimensional Quasiconformal Mappings

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*Frederick W. Gehring; Gaven J. Martin; Bruce P. Palka*

This book offers a modern, up-to-date
introduction to quasiconformal mappings from an explicitly geometric
perspective, emphasizing both the extensive developments in mapping
theory during the past few decades and the remarkable applications of
geometric function theory to other fields, including dynamical
systems, Kleinian groups, geometric topology, differential geometry,
and geometric group theory. It is a careful and detailed introduction
to the higher-dimensional theory of quasiconformal mappings from the
geometric viewpoint, based primarily on the technique of the conformal
modulus of a curve family. Notably, the final chapter describes the
application of quasiconformal mapping theory to Mostow's celebrated
rigidity theorem in its original context with all the necessary
background.

This book will be suitable as a textbook for
graduate students and researchers interested in beginning to work on
mapping theory problems or learning the basics of the geometric
approach to quasiconformal mappings. Only a basic background in
multidimensional real analysis is assumed.

#### Readership

Graduate students and researchers interested in mapping theory.

#### Reviews & Endorsements

GMP are to be commended for writing their monograph with a remarkable attention to detail...Readers keen to engage with the material could easily attempt to prove any of the results as exercises. I am confident that this book will very soon become a standard basic reference.

-- Tushar Das, MAA Reviews

The content of this book is precisely articulated while an inviting and, at times, conversational tone is maintained. This combination makes for a pleasant reading experience. The presentation of content is clearly organized and follows a natural progression of ideas.

-- David Matthew Freeman, Mathematical Reviews

[T]he book takes a wider approach to the modern theory of quasiconformal mappings and its applications than what is usual in more specialized books.

-- Olli Martio, Zentralblatt MATH

#### Table of Contents

# Table of Contents

## An Introduction to the Theory of Higher-Dimensional Quasiconformal Mappings

- Cover Cover11
- Title page i2
- Contents v6
- Preface vii8
- Chapter 1. Introduction 112
- Chapter 2. Topology and Analysis 516
- Chapter 3. Conformal Mappings in Euclidean Space 1728
- 3.1. Linear conformal transformations 1728
- 3.2. Reflections 2031
- 3.3. The Möbius group 2334
- 3.4. Hyperbolic geometry 3748
- 3.5. Classification of hyperbolic isometries 4758
- 3.6. The distortion, compactness and convergence properties of Möbius transformations 4960
- 3.7. The Möbius group as a matrix group 5667
- 3.8. Liouville’s theorem 6475

- Chapter 4. The Moduli of Curve Families 7788
- Chapter 5. Rings and Condensers 151162
- Chapter 6. Quasiconformal Mappings 205216
- 6.1. The definition of quasiconformality via conformal moduli 205216
- 6.2. Examples and the computation of dilatation 210221
- 6.3. Some measure theory 222233
- 6.4. The analytic characterisation of quasiconformality 229240
- 6.5. The boundary behavior of quasiconformal mappings 251262
- 6.6. The distortion, compactness and convergence properties of quasiconformal families 271282
- 6.7. Quasiconformal mappings of \IHⁿ with the same boundary values 298309
- 6.8. The 1-quasiconformal mappings 300311

- Chapter 7. Mapping Problems 307318
- Chapter 8. The Tukia-Väisälä Extension Theorem 355366
- Chapter 9. The Mostow Rigidity Theorem and Discrete Möbius Groups 381392
- 9.1. Introduction and statement of the theorem 381392
- 9.2. Hyperbolic manifolds, covering spaces and Möbius groups 384395
- 9.3. Quasiconformal manifolds and quasiconformal mappings 388399
- 9.4. Quasi-isometries 390401
- 9.5. Groups as geometric objects 393404
- 9.6. The boundary values are quasiconformal 398409
- 9.7. The limit set of a Möbius group 402413
- 9.8. Mappings compatible with a Möbius group 409420
- 9.9. The proof of Mostow’s theorem 412423

- Basic Notation 417428
- Bibliography 419430
- Index 427438
- Back Cover Back Cover1442